Question

Figure 6-16 shows the overhead view of the path of an amusement-park ride that travels at constant speed through five circular arcs of radii,${\mathit{R}}_{\mathbf{0}}$,$\mathbf{2}{\mathit{R}}_{\mathbf{0}}$and$\mathbf{3}{\mathit{R}}_{\mathbf{0}}$. Rank the arcs according to the magnitude of the centripetal force on a rider traveling in the arcs, greatest first.

Short answer

The rank of the arcs according to the magnitude of the centripetal force on the rider traveling in the arcs is${\mathrm{R}}_4>{\mathrm{R}}_2={\mathrm{R}}_3>{\mathrm{R}}_1={\mathrm{R}}_5$ .

Step-by-step solution

The given data

a) The rider is riding at a constant speed.

b) Arc radii from the figure for arcs 1 to 5 are:

$$\begin{gathered} {\mathrm{R}}_1=3{\mathrm{R}}_0 \\ {\mathrm{R}}_2=2{\mathrm{R}}_0 \\ {\mathrm{R}}_3=2{\mathrm{R}}_0 \\ {\mathrm{R}}_4={\mathrm{R}}_0 \\ {\mathrm{R}}_5=3{\mathrm{R}}_0 \end{gathered}$$

Understanding the concept of the centripetal force

Using the concept of centripetal acceleration with the radius of the arc, we can get the required values of the accelerations for constant speed and variable radii.

Formula:

The centripetal force due to circular motion,

$\mathbf{F}\mathbf{=}\frac{{\mathbf{mv}}^{\mathbf{2}}}{\mathbf{r}}$ (1)

Calculation of the magnitude of the centripetal force to rank the arcs

From the relation of the acceleration to the radius of the arc of equation (1), we can get that$F_c$ is inversely proportional to the radius of the curve R .

Thus the rank of the arcs according to the magnitude of the centripetal force is${\mathrm{R}}_4>{\mathrm{R}}_2={\mathrm{R}}_3>{\mathrm{R}}_1={\mathrm{R}}_5$ .